Random matrix theory and the boson peak in two-dimensional systems

The random matrix theory is used to describe the vibrational properties of two-dimensional disordered systems with a large number of degrees of freedom. It is shown that the correlated Wishart ensemble allows one to take into account the most significant mechanical properties of amorphous solids. In this ensemble, an excess density of vibrational states in comparison with the Debye law is observed, which is expressed in a peak in the reduced density of states $g(\omega)/\omega$. It is known as the boson peak, observed in many experiments and numerical calculations concerning two-dimensional and three-dimensional disordered systems. It is shown that the asymptotic behavior of the boson peak in two-dimensional systems has a number of features.